where \(\iota \)  is as in Proposition REF . The first inequality is Lemma REF  and it follows mainly from [1]}. The second inequality
is Lemma REF . That result builds on the proof of a matrix generalization of Friedman's inequality by [2]}. It requires careful technical analysis, however, which is given in Appendix REF . The challenge is that we need a relation between matrices like \(S_{T,i} \preceq \iota \tilde{S}_{T,I^*} \)  to hold with high probability, whereas [2]} bounds \(\lambda _{\max }(S_{T,i}^{-1} - \tilde{S}^{-1}_{T,i})\)  in terms of \(\lambda _{\max }(\tilde{S}^{-1}_{T,i})\) .
